# Injectivity of Laplace Transform

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## Theorem

Let $f$, $g$ be functions from $\R_{\ge 0} \to K$ of a real variable $t$, where $K \in \set {\R, \C}$.

Further let $f$ and $g$ be continuous everywhere on their domains.

Let $f$ and $g$ both admit Laplace transforms.

Suppose that the Laplace transforms $\laptrans f$ and $\laptrans g$ satisfy:

- $\forall t \in \R_{\ge 0}: \laptrans {\map f t} = \laptrans {\map g t}$

Then $f = g$ everywhere on $\R_{\ge 0}$.

### Corollary

Let $f$ and $g$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\R_{\ge 0}$.

Then $f = g$ everywhere on $\R_{\ge 0}$, except possibly where $f$ or $g$ have discontinuities of the first kind.

## Proof

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